
// Make newform 5184.2.f.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5184_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5184_f_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_5184_2_f_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5184_2_f_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, -8, 0, 49, 0, -104, 0, 160, 0, -104, 0, 49, 0, -8, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3579, 0, -12686, 0, 28832, 0, -10680, 0, 2960, 0, 3136, 0, -597, 0, 114], [2183, 0, -441, 0, 2688, 0, -10072, 0, 8232, 0, -4704, 0, 791, 0, -105, 0], [0, 3893, 0, -13719, 0, 27200, 0, -36424, 0, 22888, 0, -10112, 0, 1637, 0, -199], [0, -8927, 0, 31481, 0, -62432, 0, 85144, 0, -54248, 0, 24512, 0, -3983, 0, 489], [260, 0, -2351, 0, 6192, 0, -9538, 0, 6280, 0, -2916, 0, 476, 0, -59, 0], [0, -590, 0, 17619, 0, -26432, 0, 46256, 0, -29336, 0, 15104, 0, -2478, 0, 323], [80, 0, 63, 0, -384, 0, 1616, 0, -1176, 0, 672, 0, -112, 0, 15, 0], [0, -10894, 0, 38481, 0, -79520, 0, 83984, 0, -50392, 0, 19040, 0, -3022, 0, 337], [893, 0, -8708, 0, 16352, 0, -28616, 0, 18816, 0, -9344, 0, 1533, 0, -196, 0], [-40, 0, 441, 0, -336, 0, 944, 0, -600, 0, 384, 0, -64, 0, 9, 0], [8969, 0, -80105, 0, 221376, 0, -335752, 0, 221992, 0, -100704, 0, 16409, 0, -2009, 0], [0, 1565, 0, -27681, 0, 70112, 0, -122696, 0, 81992, 0, -40064, 0, 6573, 0, -833], [-201, 0, -336, 0, 2048, 0, -8336, 0, 6272, 0, -3584, 0, 599, 0, -80, 0], [0, 3635, 0, -63867, 0, 162848, 0, -284984, 0, 191096, 0, -93056, 0, 15267, 0, -1931], [0, -3695, 0, 71106, 0, -165536, 0, 289688, 0, -192992, 0, 94592, 0, -15519, 0, 1970]];
    Rf_basisdens := [1, 528, 1056, 528, 1056, 198, 528, 48, 528, 528, 24, 3168, 528, 88, 1056, 528];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5184_f();"
function MakeCharacter_5184_f()
    N := 5184;
    order := 2;
    char_gens := [2431, 325, 1217];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5184_f_Hecke();"
function MakeCharacter_5184_f_Hecke(Kf)
    N := 5184;
    order := 2;
    char_gens := [2431, 325, 1217];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0], [2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 1], [2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2, -2], [0, 0, 0, 0, 0, 4, 0, 0, 0, -1, 1, -1, 0, 0, 0, 0], [8, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 2, 2], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 0, 1, -1], [-2, 0, -2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 2, 4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 9, 0, 0, 0, 0, -1, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 3, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0, -1, -2], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, -1, 1], [4, 0, -4, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 2, 0, 0], [0, 1, 0, 1, -2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, 0, 0, 0, 5, 1, 1, 0, 0, 0, 0], [2, 0, -2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 3, 0, 0], [0, -2, 0, 0, -3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 3, 6, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 2, 0, 0, 2], [0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 4, -5, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [3, 0, -4, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, -3, -2, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0], [14, 0, 1, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 6, -3, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 3, 2], [0, 0, 0, 0, 0, 5, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 5, 0, -3, -3], [0, 4, 0, 2, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 3, 0, 1, 3], [0, -3, 0, 6, 2, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 5, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 8, 1, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 0, 0, 0, -8, 2, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, -4], [6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 1, 0, -7, -4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [4, 0, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 1, 0, 0], [0, -3, 0, -4, -6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -1, 0, 0], [0, 2, 0, 3, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [14, 0, 7, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -2, 0, 0], [4, 0, 4, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, -1, -4], [0, 0, 0, 0, 0, 6, 0, 0, 0, 5, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 1], [0, 5, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 0, 0, 0, 9, 1, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 0, 0, 0, 2, -2, -6, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 1, 0, 0], [0, 3, 0, 4, 9, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -4, -2, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 2, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -5, 0, -1, -3], [2, 0, 2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -4, 0, -4, 0], [0, 0, 0, 0, 0, 13, 0, 0, 0, 2, -1, -1, 0, 0, 0, 0], [0, -4, 0, 6, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [15, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, 0, -6, 0, 0, 0, -1, 3, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 2, -5], [-6, 0, -8, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 4, 2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0, 0, 3, -1, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 1, 0, -4, -3, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -18, 0, 0, 0, -13, -1, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 3, 2], [2, 0, -5, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, -1, 0, 0], [0, 5, 0, -4, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, -4, 4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, -6, -1], [-6, 0, 3, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, 0, 0, 0, -10, -1, -1, 0, 0, 0, 0], [-2, 0, 2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 1, 0, 0], [0, 2, 0, -3, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 10, 0, 2, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 3, 0, -3, 2], [0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -3, 4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 5, 3], [0, -3, 0, -2, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, -2, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, -4, 0, 0, 0, 10, -5, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, -14, 0, 0, 0, -4, -2, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, -1, 0, 1, -3], [-8, 0, -10, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 4, 0, 0, 0, -10, 4, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 5, 0, -5, 5], [-20, 0, 2, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, -2, 0, 0], [0, 4, 0, -4, -3, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 8, -6, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -6, 0, -5, -4], [0, 0, 0, 0, 0, 8, 0, 0, 0, -6, -4, 1, 0, 0, 0, 0], [-11, 0, 6, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, -2, 0, 0], [0, -4, 0, -4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -10, 0, 6, -2], [-2, 0, -1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0], [0, 6, 0, -3, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 0, 0, -2, -1, 9, 0, 0, 0, 0], [0, -9, 0, 2, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 10, 0, 0, 0, -13, -2, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -18, 0, 0, 0, -6, 2, 0, 0, 0, 0, 0], [-2, 0, -4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -4, 0, 0], [0, -4, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 8, 0, 0, 0, 7, 2, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, 0, 0, 0, -4, 4, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 9, 0, -3, -7], [9, 0, 2, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -7, 6, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [-14, 0, -1, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0, 1, 0, 0], [0, 3, 0, 9, 4, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 7, 0, -1, -3], [-22, 0, 4, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 4, 0, 0], [0, -6, 0, -3, -6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0, 0, -18, 2, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 4, 0, 3, -4], [0, 0, 0, 0, 0, -12, 0, 0, 0, 2, -2, 6, 0, 0, 0, 0], [0, -1, 0, 6, 9, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -32, 0, 0, 0, -6, -2, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 7, 0, -9, -1], [-4, 0, -2, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 1, 0, 0], [0, -8, 0, 2, -5, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, 0, 0, 0, -8, -2, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 2, 0, 8, 5], [-16, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, -6, 0, 6, 1, 0, 0, 0, -2, 0, 0, 0, 0, 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0, 0, 0, 0, 0, -2, 0, 10, -5], [18, 0, 2, 0, 0, 0, 0, -22, 0, 0, 0, 0, 0, 3, 0, 0], [36, 0, 15, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [12, 0, 9, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, -3, 0, 0], [0, -11, 0, -4, 7, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 16, 6, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 48, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0], [0, 8, 0, -5, -2, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 16, 0, 0, 0, -16, -2, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, -12, 0, 23, -4], [40, 0, -7, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 12, 0, -2, -6], [0, 0, 0, 0, 0, 26, 0, 0, 0, -29, -1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0, 5, 0, -7, 18], [22, 0, -6, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -6, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, -20, 0, 0, 0, 27, -1, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, -2, 0, -12, -7], [0, 13, 0, -7, -2, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -20, 0, 0, 0, 17, -7, 4, 0, 0, 0, 0], [0, 19, 0, -1, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -10, 0, 5, 12], [0, 0, 0, 0, 0, -7, 0, 0, 0, -1, 14, -5, 0, 0, 0, 0], [44, 0, -2, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, -2, 12, 0, 0, 0, -18, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, -5, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [32, 0, 1, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 1, 0, 0], [4, 0, 6, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 7, 0, 0], [0, 6, 0, 7, 8, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -24, 0, 0, 0, -20, -9, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 16, 0, 0, 0, -18, 10, -7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 5, 0, -11, 8], [0, 4, 0, 6, -10, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, -8, 0, -3, -8], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -10, 0, -4, 3], [0, -9, 0, -4, -17, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, -31, 9, -7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 1, 0, 15, -1], [0, 17, 0, 6, -1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -4, 0, 8, -2], [18, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 1, 0, 0], [-5, 0, -2, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 5, 0, 0], [0, -2, 0, -8, 12, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 16, 0, 0, 0, -10, -2, -6, 0, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, -38, 0, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, -9, 6], [24, 0, -3, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 2, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, -3, 10, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 16, 0, 0, 0, 23, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 41, 0, 0, 0, 21, -2, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 4, 0, -1, -2], [0, 10, 0, 4, -2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [19, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 16, 0, 2, 1], [-34, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0, 3, 0, 0], [0, -5, 0, 3, 2, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -18, 0, 0, 0, 0, 0, -1, 0, -1, 12], [0, 0, 0, 0, 0, -40, 0, 0, 0, 16, -2, 8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 15, 0, 5, -10], [18, 0, 9, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -8, 0, 0], [0, -5, 0, 6, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 14, 0, 0, 0, 12, 0, 12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, 0, 0, 0, 3, 0, 17, 0, 0, 0, 0], [8, 0, -5, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -8, 0, 0], [0, -7, 0, -12, -16, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 1, 0, 0], [0, 11, 0, 1, 10, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -12, 0, -15, -6], [46, 0, 11, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 4, 8, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 6, 0, 0, 0, 0, -18, 0, 0, 0, 0, 0, 9, 0, 0], [0, 0, 0, 0, 0, -28, 0, 0, 0, -8, 1, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, -19, 0, 3, 2], [40, 0, -7, 0, 0, 0, 0, -17, 0, 0, 0, 0, 0, 6, 0, 0], [0, 10, 0, 13, 14, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 1, -1, -13, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, -2, 0, -4, -21], [0, 0, 0, 0, 0, -14, 0, 0, 0, 18, -3, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -15, 0, 23, -6], [0, -8, 0, 1, -8, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 12, 0, 0, 0, 31, 6, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 23, 0, 0, 0, 10, 3, 19, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, 0, -9, 0], [-12, 0, -15, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -2, 0, -4, 6], [25, 0, -12, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -2, 0, 0], [0, -8, 0, -9, 12, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, -10, 0, 11, 14], [28, 0, 2, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 7, 0, 0], [-6, 0, -8, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 6, 0, 0], [0, 6, 0, 3, -8, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, -22, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -1, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, -19, 0, 7, 13], [0, 11, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 22, 16, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 54, 0, 0, 0, -15, -5, 12, 0, 0, 0, 0], [0, 0, 0, 12, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 8, -4], [-17, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, 0, -5, 0, 0], [0, 8, 0, -17, -12, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, -2, 0, 14, -16], [0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 1, 0, -3, 8], [36, 0, -12, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 9, 0, 0], [0, 0, 0, -4, -14, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5184_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5184_2_f_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_5184_2_f_e(:prec:=16)
    chi := MakeCharacter_5184_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5184_2_f_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_5184_2_f_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5184_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, 0, -216, 0, 138, 0, -24, 0, 1]>,<19,R![36, 36, -6, -6, 1]>],Snew);
    return Vf;
end function;